Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5

Average Error: 35.9 → 27.8

Time: 13.6s

Precision: 64

Math

\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 2.696758195644417099856582353822886943817:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left({\left(\left(\sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right)}^{3}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

C

double f(double x, double y) {
        double r572837 = x;
        double r572838 = y;
        double r572839 = 2.0;
        double r572840 = r572838 * r572839;
        double r572841 = r572837 / r572840;
        double r572842 = tan(r572841);
        double r572843 = sin(r572841);
        double r572844 = r572842 / r572843;
        return r572844;
}

↓

double f(double x, double y) {
        double r572845 = x;
        double r572846 = y;
        double r572847 = 2.0;
        double r572848 = r572846 * r572847;
        double r572849 = r572845 / r572848;
        double r572850 = tan(r572849);
        double r572851 = sin(r572849);
        double r572852 = r572850 / r572851;
        double r572853 = 2.696758195644417;
        bool r572854 = r572852 <= r572853;
        double r572855 = cbrt(r572852);
        double r572856 = r572855 * r572855;
        double r572857 = r572856 * r572855;
        double r572858 = 3.0;
        double r572859 = pow(r572857, r572858);
        double r572860 = pow(r572859, r572858);
        double r572861 = cbrt(r572860);
        double r572862 = cbrt(r572861);
        double r572863 = 1.0;
        double r572864 = r572854 ? r572862 : r572863;
        return r572864;
}

Error

Bits error versus x

Bits error versus y

Target

Original	35.9
Target	28.9
Herbie	27.8

\[\begin{array}{l} \mathbf{if}\;y \lt -1.230369091130699363447511617672816900781 \cdot 10^{114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \lt -9.102852406811913849731222630299032206502 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))) < 2.696758195644417

   1. Initial program 25.1

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 45.5

      \[\leadsto \frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\color{blue}{\sqrt[3]{\left(\sin \left(\frac{x}{y \cdot 2}\right) \cdot \sin \left(\frac{x}{y \cdot 2}\right)\right) \cdot \sin \left(\frac{x}{y \cdot 2}\right)}}}\]

   4. Applied add-cbrt-cube 45.1

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\tan \left(\frac{x}{y \cdot 2}\right) \cdot \tan \left(\frac{x}{y \cdot 2}\right)\right) \cdot \tan \left(\frac{x}{y \cdot 2}\right)}}}{\sqrt[3]{\left(\sin \left(\frac{x}{y \cdot 2}\right) \cdot \sin \left(\frac{x}{y \cdot 2}\right)\right) \cdot \sin \left(\frac{x}{y \cdot 2}\right)}}\]

   5. Applied cbrt-undiv 45.1

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\tan \left(\frac{x}{y \cdot 2}\right) \cdot \tan \left(\frac{x}{y \cdot 2}\right)\right) \cdot \tan \left(\frac{x}{y \cdot 2}\right)}{\left(\sin \left(\frac{x}{y \cdot 2}\right) \cdot \sin \left(\frac{x}{y \cdot 2}\right)\right) \cdot \sin \left(\frac{x}{y \cdot 2}\right)}}}\]

   6. Simplified 25.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}^{3}}}\]
   7. Using strategy rm

   8. Applied add-cbrt-cube 25.1

      \[\leadsto \sqrt[3]{\color{blue}{\sqrt[3]{\left({\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}^{3} \cdot {\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}^{3}\right) \cdot {\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}^{3}}}}\]

   9. Simplified 25.1

      \[\leadsto \sqrt[3]{\sqrt[3]{\color{blue}{{\left({\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}^{3}\right)}^{3}}}}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 25.1

       \[\leadsto \sqrt[3]{\sqrt[3]{{\left({\color{blue}{\left(\left(\sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right)}}^{3}\right)}^{3}}}\]

   if 2.696758195644417 < (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0))))

   1. Initial program 62.7

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]

   2. Taylor expanded around 0 34.5

      \[\leadsto \color{blue}{1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 27.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 2.696758195644417099856582353822886943817:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left({\left(\left(\sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right) \cdot \sqrt[3]{\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}}\right)}^{3}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< y -1.23036909113069936e114) 1 (if (< y -9.1028524068119138e-222) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1))

  (/ (tan (/ x (* y 2))) (sin (/ x (* y 2)))))